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Quantum Entanglement dependence on

bifurcations and scars in non autonomous

systems. The case of Quantum Kicked Top

George Stamatiou Demetris P.K. Ghikas ∗

Department of Physics, University of Patras,Patras 26500, Greece

February 1, 2008

Abstract

Properties related to entanglement in quantum systems, are knownto be associated with distinct properties of the corresponding classicalsystems, as for example stability, integrability and chaos. This meansthat the detailed topology, both local and global, of the classical phasespace may reveal, or influence, the entangling power of the quantumsystem. As it has been shown in the literature, the bifurcation points,in autonomous dynamical systems, play a crucial role for the onset ofentanglement. Similarly, the existence of scars among the quantumstates seems to be a factor in the dynamics of entanglement. Here westudy these issues for a non-autonomous system, the Quantum KickedTop, as a collective model of a multi-qubit system. Using the bifurca-tion diagram of the corresponding classical limit (the Classical KickedTop), we analyzed the pair-wise and the bi-partite entanglement ofthe qubits and their relation to scars, as a function of the critical pa-rameter of the system. We found that the pair-wise entanglement andpair-wise negativity show a strong maximum precisely at the bifurca-tion points, while the bi-partite entanglement changes slope at thesepoints. We have also investigated the connection between entangle-ment and the fixed points on the branch of the bifurcation diagram

∗Corresponding author: ghikas@physics.upatras.gr, Tel: +302610997460

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between the two first bifurcation points and we found that the entan-glement measures take their extreme values precisely on these points.We conjecture that our results on this behavior of entanglement isgeneric for many quantum systems with a nonlinear classical ana-logue.PACS : 03.67Mn, 05.45Ac, 05.45Pq, 05.45MtKEY WORDS : Entanglement, Quantum Chaos, Bifurcations, Scars,Kicked-Top Model.

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1. INTRODUCTION

Quantum systems with classical analogues can show properties whichstrongly depend on the detailed topological structure of the classical phasespace. One of the oldest and intensively studied properties is the quantumsignature of classical chaos, the so called quantum chaos [1]. In a broadsense , the last concept is mostly defined in relation to the classical system.Namely, knowing the parameter ranges for which the corresponding classicalsystem is chaotic, one associates quantum structures with characteristic clas-sical behavior in these parameter ranges, as for example the well establishedeigenvalue and eigenvector distributions [2, 3]. On the other hand it hasbeen a lot of research activity for a purely quantum mechanical definitionof chaos in terms of intrinsic quantum properties, the most characteristicof these being the quantum entanglement [4]. Indeed it has been shownthat the entangling power of the evolution operator and the stability of en-tanglement are strongly influenced by the properties of the classical regime[6, 9, 10, 11, 12, 7, 8, 13, 14, 15, 16, 17]. Though there are some conflict-ing results and mostly dependent on specific models, the consensus is thatquantum entanglement is an effective signature of quantum chaos. One thenmay pose the question whether the more detailed structure of the classicalphase space may reveal a corresponding structure or behavior of the quan-tum system. Thus it is natural to be asked if the bifurcation points cansignal anything for the quantum regime. After all, the most common scenar-ios of classical chaos are associated with specific series of bifurcations. Thequestion that has been analyzed in the literature is the behavior of entan-glement at the bifurcation points, and it has been shown ,for autonomoussystems, that indeed entanglement is sensitive to this classical information[18, 19, 20, 21]. In a different direction, there is an established relationshipbetween classical periodic orbits and localization of wavefunctions, the socalled scars [22, 23, 24, 25, 26, 27]. The existence of scars is closely related tothe chaotic regime, and an immediate question is whether entanglement andscars are related . There exist studies of autonomous systems which answerthis question positively [31]. In this paper we study these issues for a nonautonomous system, the Quantum Kicked Top [28, 29, 30]. We use this tomodel the collective behavior of a system of qubits perturbed externally bydiscrete kicks. This system has been studied extensively in the literature. Itis well understood how the classical and quantum chaos are interrelated, whatis the role of periodic orbits, how entanglement production may be quantified

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and analyzed with this model and finally how entanglement may be used asa signature of quantum chaos. Here first we study the role of bifurcationsin the entangling power of the kicked top. We have found that the pair-wiseentanglement shows a strong maximum precisely at the bifurcation points.Then we analyzed the relation of the existence of scars to entanglement prop-erties and we have observed that the bi-partite entanglement as a functionof the critical parameter shows a strong change of slope. Finally to have amore complete picture, we investigated the dependence of entanglement onthe position of the phase space point with respect to the branch of stablefixed points. We found that , precisely on this branch , between the first twobifurcation points, entanglement becomes extremum. Of course a completeunderstanding of this behavior would come from an extensive analysis of therelation between entanglement and the stability properties of the points onhigher branches of the bifurcation diagram, high values of k, well inside thechaotic regime. In section 2 we introduce the model and review some knownfacts from the literature. We also present the concept of quantum scars asit is applied in this model. In section 3 we present our calculations and inthe final section we discuss our results and present a conjecture about theirgenerality.

2. BACKGROUND

2.1Quantum and Classical Kicked Top

For a system of N qubits we define the collective operators

Ja =N∑

i=1

σia

2(1)

with {σia} the Pauli matrices. We use as a basis the symmetric Dicke states[33] {|j, m >, m = −j, · · · , j, j = N/2}. This means that the dimension of thespace is N+1 and not 2N , a fact that facilitates the numerical computations.We model the dynamics of this system with the Quantum Kicked Top (QKT)Hamiltonian [23]. This is the non autonomous quantum system

H(t) = (hp/τ)Jy + (hk/2j)J2z

+∞∑

n=−∞δ(t − nτ) (2)

The linear term describes a precession around the y axis with angular fre-quency p/τ . The quadratic perturbation , with perturbation parameter k

2

accounts for periodic kicks in time steps with period τ . The unitary timeevolution is given by the Floquet operator

F = e−i(k/2j) J2z e−ipJy (3)

with the normalization h = 1, τ = 1. The evolution is a precession aroundthe y axis by an angle p with a subsequent kick. If the initial state of thesystem is |y0 >, after n units of time the state is given by |yn > = F n|y0 >.The equation

F |Φm > = eixm|Φm > (4)

determines the pseudo-eigenfunctions and the pseudo-eigenvalues of the Flo-quet operator. These functions provide a complete set for the expansion ofthe states of the system, namely

|yn > =∑

m

einxm|Φm >< Φm|y0 > (5)

We note here that this dynamics leaves the set of symmetric states invariant.The transition to the corresponding Classical Kicked Top is achieved byfirst expressing the dynamics of the system in the Heisenberg Picture. Theevolution of the angular momentum operators

Jin = F †nJ0Fn (6)

gives the equations

J′

x =1

2(Jxcosp + Jzsinp + iJz)e

i kj(Jzcosp−Jxsinp+ 1

2) + h.c (7)

J′

y =1

2i(Jxcosp + Jzsinp + iJz)e

i kj(Jzcosp−Jxsinp+ 1

2) + h.c (8)

J′

z = Jzcosp − Jxsinp (9)

In order to define the corresponding Classical Kicked Top one has to takethe limit of very large angular momentum j. Defining the new operatorsTa = Ja/j , a ∈ {x, y, z} which satisfy [Ta, Tb] = iǫabc

jTc, we have that for

j → ∞ these operators commute. Then writing X = Tx, Y = Ty, Z = Tz, inthis limit , we have the classical equations of motion

X ′ = Re(Xcosp + Xsinp + iY )eik(Zcosp−Xsinp) (10)

Y ′ = Im(Xcosp + Xsinp + iY )eik(Zcosp−Xsinp) (11)

Z ′ = −Xsinp + Zcosp (12)

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For the specific value p = π/2 the dynamics is given by the equations

X ′ = Zcos(kX) + Y sin(kX) (13)

Y ′ = −Zsin(kX) + Y cos(kX) (14)

Z ′ = −X (15)

and since we have

X ′2 + Y ′2 + Z ′2 = X2 + Y 2 + Z2 = 1 (16)

the transformation X = sinθcosφ , Y = sinθsinφ , Z = cosθ, gives a twodimensional map as the equations of motion for the Classical Kicked Top.This classical dynamics is well analyzed in the literature [28, 29, 30]. Forsmall values of k we have islands of regularity which are destroyed as k in-creases. For intermediate values of k we have a mixed phase space, while forbig values we have full chaos. There are two trivial equilibrium points, atthe north and south poles of the sphere, defining these points here w.r.t. Yaxis. We follow the bifurcation diagram of one of these equilibrium pointsanalyzing the behavior of the corresponding quantum system at the bifurca-tion point. In a more complete analysis , we plan to track the signatures ofmany bifurcations points and picture the overall significance of the bifurca-tion landscape for the quantum behavior. There are generic scenarios of theonset of chaos, and these are well studied and understood, as well as casesof systems with singular behavior. We consider the study of the quantumsignature of these classical portraits as very important. In this paper we areonly taking the very first step of such an analysis.

2.2Quantum Scars.

In the semi-classical approximation theory of Quantum Mechanics theclassical periodic orbits play a fundamental role [1]. It has been provedthat the fluctuating part of the density of states can be written as a sumof terms over the classical periodic orbits, and that the existence of theseorbits influences the energy spectrum of the corresponding quantum system.But one of the most impressive and unexpected results is the existence ofscars [22]. These are states which are strongly localized in the phase spaceat points which are directly related to the classical periodic orbits. TheseQuantum Scars (QS) do not exist for every periodic orbit and not everyquantum state is related to any periodic orbit. But it is interesting to inves-tigate how the existence of QS affects the quantum properties of the system.

For non-autonomous systems, like the QKT, scars can be found among thepseudo-eigenstates of the Floquet operator. It has been proved that scarsplay a significant role in the time evolution of a coherent state which startsfrom a point of the periodic orbit [22, 23, 24, 25, 26, 27]. In this paper weanalyze the relation of scars to the entanglement. We base our analysis onsome known results.

Let |γ >= |θ, φ > be a given state of the QKT where |θ, φ > is the spincoherent state defined by [32], {|θ, φ >= R(θ, φ)|j, j >;−π ≤ φ ≤ π, 0 ≤ θ ≤π}, with

R(θ, φ) = exp{iθ[Jxsinφ − Jycosφ]} (17)

Then for the nth iterate of the Floquet operator we have |γ′ >= F n|γ >.Here γ′ is the coherent state parameter given by the classical dynamics γ′ =T (· · ·T (T (γ)))︸ ︷︷ ︸

n

, where T is the appropriate classical limit of the Floquet map.

Thus for a classical periodic orbit of order K we have γ0 = T (· · ·T (T (γ0)))︸ ︷︷ ︸

K

,

[28]. Then for the M-step propagator fM =< γ0|FM |γ0 > we have, after

expanding in the pseudo-eigenfucntion basis

fM =N∑

n=1

Hn(γ0)eiMxn (18)

where N is the dimensionality of the system and Hn = | < Φn|γ0 > |2

the Husimi representation [23] of the nth pseudo-eigenfunction Φn. Theimportance of this quantity is that its Fourier Transform

fω =L∑

M=−L

e−iωMfM =N∑

n=1

Hn(γ0)L∑

M=−L

eiM(φn−ω) (19)

contains the information for the quantum scars. That is, when ω is closeto a pseudo-eigenenergy, the spectral density is dominated by the overlapfunction of the corresponding pseudo-eigenfunction if the latter is associatedto a scar. Thus we locate the QS that corresponds to the periodic orbitΓ = (γ0, γ1, · · · , γM−1) as a maximum in fω. Stated differently, the eigen-function associated with the energy at the peak is scarred by this periodicorbit [23]. We refer to the literature for more details on scars and their prop-erties.

2.3 Entangling Measures.

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Our main objective is to study the pair-wise and the bi-partite entagle-ment in a system of N qubits which is subjected to a time-dependent externalperturbation and analyze the interrelation of these quantities with the ex-istence of bifurcation points of the corresponding classical system and therole of quantum scars. As we said above, we model this collective behavioras Classical and Quantum Kicked Tops, the properties of which are very ex-tensively analyzed in the literature. We quantify the pair-wise entanglementwith both the entanglement of formation and negativity and the bi-partiteentanglement with the linear entropy. More specifically, let ρ be the densitymatrix of the system and |i1i2 · · · iN > ik ∈ {0, 1} the computational basis.Let ρk = trALL6=k(ρ) and ρkm = trALL6=k,m(ρ) be the one qubit and two qubitsreduced density matrices. We use three measures of entanglement. The bi-partite entanglement Q ≡ 2− 2

N

∑

i tr(ρ2i ), the Negativity Np = N(ρkm) and

the Entropy of Formation Ep = Eformation(ρkm). Negativity is defined as

follows : For a generally mixed state ρ, N(ρ) =∑

j max{0,−µj} where {µj}are the eigenvalues of the partial transpose of ρ. Entanglement of Formation

is defined as Ef = H(1+√

1−C2

2) where H(x) is the Shannon entropy function

and C the concurrence defined by C = max{0, λ1 − λ2 − λ3 − λ4} and thequantities λj are , in decreasing order , the square roots of the eigenvalues ofthe matrix ρ12(σ1y ⊗ σ2y)ρ

∗12(σ1y ⊗ σ2y)). We refer to [36, 37, 38, 41, 39, 40]

for more details on the properties of these quantities. In the Discussion wecomment on the use of Generalized Entanglement [34, 35, 17].

3. THE RESULTS

3.1The Classical Picture.

We first present a series of portraits of the Classical Kicked Top for differentvalues of the parameter k and the bifurcation diagrams for the angles θ0 andφ0 of the equilibrium point of the south pole.

In Figure 1 we reproduce the phase space portrait for four different valuesof the coupling constant parameter k, with one value at the bifurcation pointk=2. We present these four figures to point to the way the classical phasespace picture is evolving with the increase of parameter k. Since we are notstudying here the classical behavior of this system, we are not concerned withthe small structures evident in the figures as we approach the chaotic regime.It is , of course, very interesting to study the relation of these details to thequantal properties of the associated system. We only note here, that in the

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limit of fully developed chaos, there exist in general unstable periodic orbitswhich we expect to play a role in the behavior of the quantum system. InFigure 2 we plot the bifurcation diagrams for the angles θ and φ which startfrom the equilibrium point in the south pole. We note that the fuzziness ofthe bifurcation diagrams after the point k=2 and before the area of completechaos is due to numerical inaccuracies. In this figure, the first bifurcationpoint is more than evident. In our results , below, for the entanglement,the signature of the existence of this point in the behavior of the quantumsystem is clearly seen. But it is not yet clear to us how to pin point the onsetof the band. Perhaps, as it is commented in the Discussion, other measures,like Generelaized Entanglement may offer a better picture.

3.2Bifurcation Points and Entanglement of Scars.

As it was described in the Introduction, scars are quantum states whichare associated to classical periodic orbits and appear as prominent structuresin phase space through their Husimi representation. In order to relate theentanglement of the scars to the bifurcation point at k=2, we first identifythe scar located at the south pole for each value of k, from k=0 to k=6.8.Then for each value of k we compute for the corresponding scar the pairwisenegativity, the pairwise entanglement of formation and the bipartite entan-glement. We did the calculations for j=35, that is N=70 qubits. In Figure3 we plot the scars and the associated spectrum corresponding to the valuesof k in Figure 1. In this figure it is seeing that as k increases more peaksare present, and as k approaches the region of full chaos the peaks multiplygreatly. We identify the highest peak with the scar that we study. The cor-responding plot of the Husimi function shows a progressive delocalization.In Figure 4 we plot the three different entanglement measures, pair-wisenegativity, pairwise entanglement and bipartite entanglement for N=50 andN=110 qubits. First we observe that in all plots the signature of the bifur-cation point at k=2 is more than evident. Second, there is not any dramaticchange with the increase of the number of qubits. In this figure we see thatthough the entanglement is associated clearly with the scarred function atthe bifurcation point, this association is not strict, and it shows a relativelywide span of values where the entanglement is appreciable. It seems thatthis is related to the progressive delocalization of the scarred state.

3.3 Entanglement and Periodic Orbits.

In Figure 5 we plot the pair-wise entanglement of formation and the bi-partite

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entanglement for time evolved states. Specifically, we start with two coher-ent states located at the south and north poles respectively, and we computetheir entanglement after 500 kicks. We repeat this for a large number of kvalues from k=0 to k=6.8. We observe again the trace of the classical bifur-cation point at k=2. There is not any difference between south and northpoles and not a strong dependence on the number of qubits for N in theorder of 10. It is interesting to note that classically the two poles behavedifferently at the bifurcation points [28]. Here , again, though the value ofk=2 is evidently seeing to be prominent, there is not a sharp peak. It seemsthat the bifurcation point is smoothly felt by the periodic orbit starting fromthe equilibrium points. A more extensive study is probably needed to under-stand the influence of the local structure of the classical phase space on thequantum behavior.

3.4 Entanglement along the branch of stable points.

To have a more complete picture of the role of the classical stable fixedpoints we analyzed the dependence of entanglement on the location of thefixed points w.r.t. the branch of the stable points between the values k=2and k=12.73, the latter corresponding to the second bifurcation point. Wecomputed the bipartite entanglement and the pairwise negativity for vari-ous values of k in the above interval. For the k=12.73 the fixed point is(θ, φ) = (2.32, 0.38), and we computed the dependence of entanglement on θand φ. We found that, at precisely this fixed point, the entanglemnt is min-imum. In Figure 6 third row, we plot the bipartite entanglement vs either θor φ. In the same figure, first and second rows we plot the results for k=2.2and k=2.4. For these cases the corresponding values for the fixed points are(θ, φ) = (1.9, 1.2), (2.1, 1.0). We keep θ fixed and we plot both the bipartiteentanglement and the pairwise negativity vs φ. Again we observe that thefixed points give minimum entanglement. We note that the same behaviouris observed for all values of k in the interval [2,12.73].

4.DISCUSSION

In this paper we posed the question of the relation of the local struc-tures of classical phase space to the purely quantum mechanical property ofentanglement. Generally, it is well established that, both qualitatively andquantitatively, entanglement and its fragility, namely the properties of thedecoherence process, strongly depend on the classical regime. That is, thepresence of classical structures, in the corresponding classical system, can be

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traced in the properties of the quantum system. But the onset of classicalchaos is generically related to bifurcations from equilibrium points, and thescenarios of chaos are pictured in connection to local or global changes ofthe phase space portrait of the system. It is generally expected that a globalchange of the classical landscape can be felt by the quantum system. But itseems very natural to ask whether a local property, that is a single bifurca-tion point can be traced in the quantum properties. It is this question thatwe have tried to answer here using scars and periodic orbits as ”tools” ofthe analysis. We have used the Classical and Quantum Kicked Tops becausethey have been well studied and understood. We use the top as a collectivemodel of N qubits. We applied three measures of entanglement , pair-wisenegativity, pair-wise entanglement of formation and bi-partite entanglement.We traced the behavior of the system for many values of the coupling pa-rameter k, from k=0 to k=6.8. As it is known for k=2 we have the firstbifurcation point of the equilibrioum at the south and north poles. First wepresented four portraits of the phase space for characteristic k values and thebifurcation diagrams of the south pole. Then for each value of k in the aboverange we locate the scarring state and compute the entanglement. The scarsalong with the associated spectra are plotted for the selected four values ofk. Finally we propagated two coherent states , initially located at the southand north poles and we computed their entanglement after 500 kicks. In allplots of the measures of entaglement the bifurcation point at k=2 is morethan evident. We conjecture that this is not a non-generic behavior. That iswe expect that the bifurcation points are intimately related to the quantumstructures. But , as it happens in the case of scars, we cannot tell whether allbifurcation points play this role and moreover whether the sensitivity of en-tanglement can always be related to a bifurcation phenomenon in the classicalregime. To establish a more complete picture, at this stage of our investiga-tion, we made an analysis of the dependence of entanglement on the properyof stability of the fixed points. On the branch of stable fixed points in thebifurcation diagram between k=2 and k=12.73 (first and second bifurcationpoints), for various values of k, we computed the dependence of entanglementon the position of the point in phase space. We observed that when the pointcrosses the branch the entanglement becomes minimum. We ploted the casesfor k=2.2, 2.4 and 12.73. The behavior is similar for other values of k in thisinterval. Given the fact that the bifurcation diagram is very complicatedit would be interesting to see whether for very high values of k, we have anew pattern on the entanglement dependence. We are currently investigating

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these issues. We end with few comments on the use of Generalized Entan-glement (GE) [34, 35, 17]. The static or dynamic signatures of quantumchaos, like eigenvalue distributions or fidelity, and entanglement generation,with the associate entanglement power of the evolution operator are widelyused, but depend strongly on the basis employed. But more importantly,they depend on the partition of the given system into subsystems, or the em-bedding into a larger system. It is this practice which we follow here usingthe pair-wise and the bi-partite entanglent and the negativity. On the otherhand the GE depends on a selected subalgebra of observables. The regimeis characterized by sums of averages of these operators. These averages aresensitive detectors of the global classical phase space, and for this reason GEis considered presently as the most effective signature of quantum chaos. Inthe present paper we were interested on two critical details of the classicaldynamics, namely the role of bifurcation points and of general periodic solu-tions in the entanglement affected by the dynamics on the subystems. Thusfrom the start our interest is on subsystems and their combined state, andaccording to [35] our bi-partite entanglement measure is actually related toGE. Of course we consider the question of the dependence of GE on the localproperties of the phase space as very important, and we are looking into thisissue.

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